Carcass wrote:
The area of circle O is added to its diameter. If the circumference of circle O is then subtracted from this total, the result is 4. What is the radius of circle O?
A) –2/pi
B) 2
C) 3
D) 4
E) 5
Let r = radius of circle
Area of circle = πr²
Diameter = 2r
Circumference of circle = 2rπ
The area of circle O is added to its diameter...We get: πr² + 2r
...If the circumference of circle O is then subtracted from this total, the result is 4We get: πr² + 2r - 2rπ = 4
From here, we CAN just
test the answer choices.
First, we can SKIP answer choice A, since the radius cannot have a negative value.
What about B (
2)?
Replace r with (
2) to get: π(
2)² + 2(
2) - 2(
2)π = 4
Simplify: 4π + 4 - 4π = 4
Simplify again: 4 = 4
PERFECT!
Answer: B
ALTERNATE SOLUTIONOnce we get the equation πr² + 2r - 2rπ = 4, we can also solve it algebraically.
To do so, we're going to
factor the expression in parts Here's what I mean...
We have: πr² + 2r - 2rπ = 4
Subtract 4 from both sides to get: πr² + 2r - 2rπ - 4 = 0
Rearrange terms to get:
πr² - 2rπ +
2r - 4 = 0
Factor as follows:
πr(r - 2) +
2(r - 2) = 0
Notice that we have (x-2) in common in both parts.
So, we can combine the parts to get: (πr + 2)(r - 2) = 0
This means that EITHER πr + 2 = 0 OR r - 2 = 0
case a: πr + 2 = 0
This means: πr = -2
Since π and r are both POSITIVE, this equation has NO SOLUTION
case b: r - 2 = 0
This tells us that r = 2
Answer: B